Simplifying fractions
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Introduction
Fractions involving symbols occur frequently. It is necessary to be able to simplify these and rewrite
them in different but equivalent forms. On this leaflet we revise how these processes are carried out.
It will be helpful if you have already seen leaflet Fractions.
Expressing a fraction in its simplest form
An algebraic fraction can always be expressed in different, yet equivalent forms. A fraction is
expressed in its simplest form by cancelling any factors which are common to both the numerator
and the denominator. You need to remember that factors are multiplied together.
For example, the two fractions
7
7a
and
ab
b
are equivalent. Note that there is a common factor of a in the numerator and the denominator of
7a
7
which can be cancelled to give .
ab
b
To express a fraction in its simplest form, any factors which are common to both the numerator
and the denominator are cancelled
Notice that cancelling is equivalent to dividing the top and the bottom by the common factor.
7
7a
It is also important to note that can be converted back to the equivalent fraction
by multiplying
b
ab
7
both the numerator and denominator of by a.
b
A fraction is expressed in an equivalent form by multiplying both top and bottom by the same
quantity, or dividing top and bottom by the same quantity
Example
The two fractions
are equivalent. Note that
10y 2
15y 5
and
2
3y 3
2×5×y×y
10y 2
=
5
15y
3×5×y×y×y×y×y
and so there are common factors of 5 and y × y. These can be cancelled to leave
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Example
The fractions
(x − 1)(x + 3)
(x + 3)(x + 5)
(x − 1)
(x + 5)
and
are equivalent. In the first fraction, the common factor (x + 3) can be cancelled.
Example
The fractions
2a(3a − b)
7a(a + b)
2(3a − b)
7(a + b)
and
are equivalent. In the first fraction, the common factor a can be cancelled. Nothing else can be
cancelled.
Example
In the fraction
a−b
a+b
there are no common factors which can be cancelled. Neither a nor b is a factor of the numerator.
Neither a nor b is a factor of the denominator.
Example
5x
as an equivalent fraction with denominator (2x + 1)(x − 7).
2x + 1
Solution
Express
To achieve the required denominator we must multiply both top and bottom by (x − 7). That is
(5x)(x − 7)
5x
=
2x + 1
(2x + 1)(x − 7)
If we wished, the brackets could now be removed to write the fraction as
5x2 − 35x
.
2x2 − 13x − 7
Exercises
1. Express each of the following fractions in its simplest form:
a)
12xy
,
16x
b)
14ab
,
21a2 b2
c)
3x2 y
,
6x
d)
3(x+1)
,
(x+1)2
e)
(x+3)(x+1)
,
(x+2)(x+3)
f)
100x
,
45
g)
a+b
.
ab
Answers
2
3
1. a) 3y
, b) 3ab
, c) xy
, d) x+1
, e) x+1
, f) 20x
, g) cannot be simplified. Whilst both a
4
2
x+2
9
and b are factors of the denominator, neither a nor b is a factor of the numerator.
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